Knot categorification from homological mirror symmetry
Knot categorification from homological mirror symmetry
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Quantum invariants of knots have many applications in mathematics and in physics. Khovanov showedÌý in ‘99 that the simplest such invariant, the Jones polynomial, arizes as the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their meaning:Ìýwhat are they homologies of?
Mirror symmetry is another important strand in the interaction between mathematics and physics. Homological mirror symmetry, formulated by Kontsevich in ’94, naturally produces hosts of homological invariants.ÌýSometimes, it can be made manifest, and then its striking mathematical power comes to fore. Typically though, it leads to invariants which have no particular interest outside of the problem at hand.
I showed recently that there is a vast new family of mirror pairs of manifolds, for which homological mirror symmetry can be made manifest.ÌýThe symplectic geometry side of mirror symmetry involves a new theory generalizing Heegard-Floer theory. They do lead to interestingÌýinvariants: in particular, they solve the knot categorification problem.